Show that the 1,2,3 2,3,1 3,1,2from a epuaiatrial triangle
Answers
Answer:
The given points form an equilateral triangle.
The proof has been shown below.
Step-by-step explanation:
Let the given points are
A(1,2,3), B(2,3,1) and C(3,1,2)
An equilateral triangle has all three sides of equal length.
Hence, for equilateral triangle,
AB = BC = CA
The distance formula is given by
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}d=
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
+(z
2
−z
1
)
2
Now, find the length of all three sides and check if they are equal or not.
\begin{gathered}AB=\sqrt{(2-1)^2+(3-2)^2+(1-3)^2}\\\\AB=\sqrt{1+1+4}\\\\AB=\sqrt{6}\end{gathered}
AB=
(2−1)
2
+(3−2)
2
+(1−3)
2
AB=
1+1+4
AB=
6
Now, find the length of side BC
\begin{gathered}BC=\sqrt{(3-2)^2+(1-3)^2+(2-1)^2}\\\\BC=\sqrt{1+4+1}\\\\BC=\sqrt{6}\end{gathered}
BC=
(3−2)
2
+(1−3)
2
+(2−1)
2
BC=
1+4+1
BC=
6
Finally, find the length of side AC
\begin{gathered}AC=\sqrt{(3-1)^2+(1-2)^2+(2-3)^2}\\\\AC=\sqrt{4+1+1}\\\\AC=\sqrt{6}\end{gathered}
AC=
(3−1)
2
+(1−2)
2
+(2−3)
2
AC=
4+1+1
AC=
6
Since, the length of all three sides are equal.
Hence, the given points form an equilateral triangle.
#Learn More:
The distance of the point p(1,2,3) from coordinate axes
Step-by-step explanation:
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